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Properties

Label 3332.1.o.f
Level $3332$
Weight $1$
Character orbit 3332.o
Analytic conductor $1.663$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -68
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(67,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.67");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 476)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.188737808.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{2} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{3} + \zeta_{12}^{4} q^{4} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{6} - q^{8} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{2} q^{2} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{3} + \zeta_{12}^{4} q^{4} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{6} - q^{8} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{9} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{11} + (\zeta_{12}^{3} + \zeta_{12}) q^{12} - q^{13} - \zeta_{12}^{2} q^{16} - \zeta_{12}^{4} q^{17} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} + 1) q^{18} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{22} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{24} + \zeta_{12}^{4} q^{25} - \zeta_{12}^{2} q^{26} + (\zeta_{12}^{5} - \zeta_{12}) q^{27} - \zeta_{12}^{4} q^{32} + ( - \zeta_{12}^{4} - 2 \zeta_{12}^{2} - 1) q^{33} + q^{34} + ( - \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{36} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{39} + (\zeta_{12}^{3} + \zeta_{12}) q^{44} + (\zeta_{12}^{5} - \zeta_{12}) q^{48} - q^{50} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{51} - \zeta_{12}^{4} q^{52} - \zeta_{12}^{4} q^{53} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{54} + q^{64} + ( - 2 \zeta_{12}^{4} - \zeta_{12}^{2} + 1) q^{66} + \zeta_{12}^{2} q^{68} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{71} + (\zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{72} + (\zeta_{12}^{3} + \zeta_{12}) q^{75} + (\zeta_{12}^{5} - \zeta_{12}) q^{78} + (\zeta_{12}^{3} + \zeta_{12}) q^{79} + \zeta_{12}^{4} q^{81} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{88} + \zeta_{12}^{2} q^{89} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{96} + (2 \zeta_{12}^{5} - 2 \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 4 q^{9} - 4 q^{13} - 2 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{25} - 2 q^{26} + 2 q^{32} - 6 q^{33} + 4 q^{34} + 8 q^{36} - 4 q^{50} + 2 q^{52} + 2 q^{53} + 4 q^{64} + 6 q^{66} + 2 q^{68} + 4 q^{72} - 2 q^{81} + 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
0.500000 + 0.866025i −0.866025 + 1.50000i −0.500000 + 0.866025i 0 −1.73205 0 −1.00000 −1.00000 − 1.73205i 0
67.2 0.500000 + 0.866025i 0.866025 − 1.50000i −0.500000 + 0.866025i 0 1.73205 0 −1.00000 −1.00000 − 1.73205i 0
2039.1 0.500000 − 0.866025i −0.866025 − 1.50000i −0.500000 − 0.866025i 0 −1.73205 0 −1.00000 −1.00000 + 1.73205i 0
2039.2 0.500000 − 0.866025i 0.866025 + 1.50000i −0.500000 − 0.866025i 0 1.73205 0 −1.00000 −1.00000 + 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
4.b odd 2 1 inner
7.c even 3 1 inner
17.b even 2 1 inner
28.g odd 6 1 inner
119.j even 6 1 inner
476.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.o.f 4
4.b odd 2 1 inner 3332.1.o.f 4
7.b odd 2 1 476.1.o.c ✓ 4
7.c even 3 1 3332.1.g.f 2
7.c even 3 1 inner 3332.1.o.f 4
7.d odd 6 1 476.1.o.c ✓ 4
7.d odd 6 1 3332.1.g.g 2
17.b even 2 1 inner 3332.1.o.f 4
28.d even 2 1 476.1.o.c ✓ 4
28.f even 6 1 476.1.o.c ✓ 4
28.f even 6 1 3332.1.g.g 2
28.g odd 6 1 3332.1.g.f 2
28.g odd 6 1 inner 3332.1.o.f 4
68.d odd 2 1 CM 3332.1.o.f 4
119.d odd 2 1 476.1.o.c ✓ 4
119.h odd 6 1 476.1.o.c ✓ 4
119.h odd 6 1 3332.1.g.g 2
119.j even 6 1 3332.1.g.f 2
119.j even 6 1 inner 3332.1.o.f 4
476.e even 2 1 476.1.o.c ✓ 4
476.o odd 6 1 3332.1.g.f 2
476.o odd 6 1 inner 3332.1.o.f 4
476.q even 6 1 476.1.o.c ✓ 4
476.q even 6 1 3332.1.g.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.1.o.c ✓ 4 7.b odd 2 1
476.1.o.c ✓ 4 7.d odd 6 1
476.1.o.c ✓ 4 28.d even 2 1
476.1.o.c ✓ 4 28.f even 6 1
476.1.o.c ✓ 4 119.d odd 2 1
476.1.o.c ✓ 4 119.h odd 6 1
476.1.o.c ✓ 4 476.e even 2 1
476.1.o.c ✓ 4 476.q even 6 1
3332.1.g.f 2 7.c even 3 1
3332.1.g.f 2 28.g odd 6 1
3332.1.g.f 2 119.j even 6 1
3332.1.g.f 2 476.o odd 6 1
3332.1.g.g 2 7.d odd 6 1
3332.1.g.g 2 28.f even 6 1
3332.1.g.g 2 119.h odd 6 1
3332.1.g.g 2 476.q even 6 1
3332.1.o.f 4 1.a even 1 1 trivial
3332.1.o.f 4 4.b odd 2 1 inner
3332.1.o.f 4 7.c even 3 1 inner
3332.1.o.f 4 17.b even 2 1 inner
3332.1.o.f 4 28.g odd 6 1 inner
3332.1.o.f 4 68.d odd 2 1 CM
3332.1.o.f 4 119.j even 6 1 inner
3332.1.o.f 4 476.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3}^{4} + 3T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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